On Rice's formula for stationary multivariate piecewise smooth processes
K. Borovkov and G. Last
Source: J. Appl. Probab. Volume 49, Number 2
(2012), 351-363.
Abstract
Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.
First Page:
Show
Hide
Keywords: Level crossing; Rice's formula; stationarity; Palm probability; piecewise-deterministic process; stochastic network
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1339878791
Digital Object Identifier: doi:10.1239/jap/1339878791
Zentralblatt MATH identifier: 06053716
Mathematical Reviews number (MathSciNet): MR2977800
References
Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
Mathematical Reviews (MathSciNet): MR2319516
Zentralblatt MATH: 1149.60003
Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.
Mathematical Reviews (MathSciNet): MR2478201
Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1288301
Bar-David, I. and Nemirovsky, A. (1972). Level crossings of nondifferentiable shot processes. IEEE Trans. Inf. Theory 18, 27–34.
Mathematical Reviews (MathSciNet): MR378073
Digital Object Identifier: doi:10.1109/TIT.1972.1054748
Borovkov, K. and Bebbington, M. S. (2003). A stochastic two-node stress transfer model reproducing Omori's law. Pure Appl. Geophys. 160, 1429–1445.
Borovkov, K. and Last, G. (2008). On level crossings for a general class of piecewise-deterministic Markov processes. Adv. Appl. Prob. 40, 815–834.
Mathematical Reviews (MathSciNet): MR2454034
Zentralblatt MATH: 1177.60067
Digital Object Identifier: doi:10.1239/aap/1222868187
Project Euclid: euclid.aap/1222868187
Borovkov, K. and Vere-Jones, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appl. Prob. 37, 315–321.
Mathematical Reviews (MathSciNet): MR1780992
Zentralblatt MATH: 0967.60077
Project Euclid: euclid.jap/1014842538
Hartman, P. (2002). Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia, PA.
Mathematical Reviews (MathSciNet): MR1929104
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
Mathematical Reviews (MathSciNet): MR1876169
Leadbetter, M. R. and Spaniolo, G. V. (2004). Reflections on Rice's formulae for level crossings–-history, extensions and use. Austral. N. Z. J. Statist. 46, 173–180.
Mathematical Reviews (MathSciNet): MR2060961
Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Tech. J. 24, 282–332.
Mathematical Reviews (MathSciNet): MR10932
Rychlik, I. (2000). On some reliability applications of Rice's formula for the intensity of level crossings. Extremes 3, 331–348.
Mathematical Reviews (MathSciNet): MR1870462
Digital Object Identifier: doi:10.1023/A:1017942408501
Zähle, U. (1984). A general Rice formula, Palm measures, and horizontal-window conditioning for random fields. Stoch. Process. Appl. 17, 265-283.
Mathematical Reviews (MathSciNet): MR751206
Zentralblatt MATH: 0541.60049
Digital Object Identifier: doi:10.1016/0304-4149(84)90005-X
Journal of Applied Probability
